Given a measurable function \(p(.):\mathbb {R}^{n}\rightarrow \left( 0,\infty \right) \)  such that \(0<p_{-}\le p_{+}<\infty ,\)  in the paper [1]} E,
Nakai and Y. Sawano give a variety of distinct approaches, based on
differing definitions, all lead to the same notion of the Hardy space \(H^{p(.)}.\)
